3.2438 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{9}{200} \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{5 x+3}}+\frac{651 \sqrt{5 x+3} (1-2 x)^{5/2}}{22000}+\frac{651 \sqrt{5 x+3} (1-2 x)^{3/2}}{8000}+\frac{21483 \sqrt{5 x+3} \sqrt{1-2 x}}{80000}+\frac{236313 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(7/2))/(275*Sqrt[3 + 5*x]) + (21483*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/80000 + (651*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/8000 + (651*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/22000 - (9*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/200 + (23631
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80000*Sqrt[10])

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Rubi [A]  time = 0.0399596, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ -\frac{9}{200} \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{5 x+3}}+\frac{651 \sqrt{5 x+3} (1-2 x)^{5/2}}{22000}+\frac{651 \sqrt{5 x+3} (1-2 x)^{3/2}}{8000}+\frac{21483 \sqrt{5 x+3} \sqrt{1-2 x}}{80000}+\frac{236313 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^(3/2),x]

[Out]

(-2*(1 - 2*x)^(7/2))/(275*Sqrt[3 + 5*x]) + (21483*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/80000 + (651*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/8000 + (651*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/22000 - (9*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/200 + (23631
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80000*Sqrt[10])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}+\frac{2}{275} \int \frac{(1-2 x)^{5/2} \left (\frac{351}{2}+\frac{495 x}{2}\right )}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}-\frac{9}{200} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{1953 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{4400}\\ &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}+\frac{651 (1-2 x)^{5/2} \sqrt{3+5 x}}{22000}-\frac{9}{200} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{651}{800} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}+\frac{651 (1-2 x)^{3/2} \sqrt{3+5 x}}{8000}+\frac{651 (1-2 x)^{5/2} \sqrt{3+5 x}}{22000}-\frac{9}{200} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{21483 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{16000}\\ &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}+\frac{21483 \sqrt{1-2 x} \sqrt{3+5 x}}{80000}+\frac{651 (1-2 x)^{3/2} \sqrt{3+5 x}}{8000}+\frac{651 (1-2 x)^{5/2} \sqrt{3+5 x}}{22000}-\frac{9}{200} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{236313 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{160000}\\ &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}+\frac{21483 \sqrt{1-2 x} \sqrt{3+5 x}}{80000}+\frac{651 (1-2 x)^{3/2} \sqrt{3+5 x}}{8000}+\frac{651 (1-2 x)^{5/2} \sqrt{3+5 x}}{22000}-\frac{9}{200} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{236313 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{80000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{3+5 x}}+\frac{21483 \sqrt{1-2 x} \sqrt{3+5 x}}{80000}+\frac{651 (1-2 x)^{3/2} \sqrt{3+5 x}}{8000}+\frac{651 (1-2 x)^{5/2} \sqrt{3+5 x}}{22000}-\frac{9}{200} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{236313 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{80000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0355591, size = 88, normalized size = 0.64 \[ \frac{-10 \left (288000 x^5-299200 x^4-147640 x^3+381870 x^2+24773 x-79699\right )-236313 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{800000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^(3/2),x]

[Out]

(-10*(-79699 + 24773*x + 381870*x^2 - 147640*x^3 - 299200*x^4 + 288000*x^5) - 236313*Sqrt[10 - 20*x]*Sqrt[3 +
5*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(800000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.011, size = 133, normalized size = 1. \begin{align*}{\frac{1}{1600000} \left ( 2880000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1552000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1181565\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-2252400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+708939\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2692500\,x\sqrt{-10\,{x}^{2}-x+3}+1593980\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^(3/2),x)

[Out]

1/1600000*(2880000*x^4*(-10*x^2-x+3)^(1/2)-1552000*x^3*(-10*x^2-x+3)^(1/2)+1181565*10^(1/2)*arcsin(20/11*x+1/1
1)*x-2252400*x^2*(-10*x^2-x+3)^(1/2)+708939*10^(1/2)*arcsin(20/11*x+1/11)+2692500*x*(-10*x^2-x+3)^(1/2)+159398
0*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 4.31685, size = 147, normalized size = 1.07 \begin{align*} -\frac{18 \, x^{5}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{187 \, x^{4}}{50 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3691 \, x^{3}}{2000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{38187 \, x^{2}}{8000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{236313}{1600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{24773 \, x}{80000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{79699}{80000 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-18/5*x^5/sqrt(-10*x^2 - x + 3) + 187/50*x^4/sqrt(-10*x^2 - x + 3) + 3691/2000*x^3/sqrt(-10*x^2 - x + 3) - 381
87/8000*x^2/sqrt(-10*x^2 - x + 3) - 236313/1600000*sqrt(10)*arcsin(-20/11*x - 1/11) - 24773/80000*x/sqrt(-10*x
^2 - x + 3) + 79699/80000/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.80111, size = 297, normalized size = 2.15 \begin{align*} -\frac{236313 \, \sqrt{10}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \,{\left (144000 \, x^{4} - 77600 \, x^{3} - 112620 \, x^{2} + 134625 \, x + 79699\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1600000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/1600000*(236313*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) - 20*(144000*x^4 - 77600*x^3 - 112620*x^2 + 134625*x + 79699)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(2+3*x)**2/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 2.20331, size = 185, normalized size = 1.34 \begin{align*} \frac{1}{2000000} \,{\left (4 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 529 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16905 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 61545 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{236313}{800000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{31250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{15625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

1/2000000*(4*(8*(36*sqrt(5)*(5*x + 3) - 529*sqrt(5))*(5*x + 3) + 16905*sqrt(5))*(5*x + 3) + 61545*sqrt(5))*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 236313/800000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/31250*sqrt(10)*(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/15625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))